Question: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $y \neq 0$. $z = \dfrac{10y - 70}{-10} \div \dfrac{y - 7}{2y} $
Solution: Dividing by an expression is the same as multiplying by its inverse. $z = \dfrac{10y - 70}{-10} \times \dfrac{2y}{y - 7} $ When multiplying fractions, we multiply the numerators and the denominators. $z = \dfrac{ (10y - 70) \times 2y } { -10 \times (y - 7) } $ $ z = \dfrac {2y \times 10(y - 7)} {-10 (y - 7)} $ $ z = \dfrac{20y(y - 7)}{-10(y - 7)} $ We can cancel the $y - 7$ so long as $y - 7 \neq 0$ Therefore $y \neq 7$ $z = \dfrac{20y \cancel{(y - 7})}{-10 \cancel{(y - 7)}} = -\dfrac{20y}{10} = -2y $